Interpretive Summary: Test day models provide more accurate estimation of genetic effects in dairy cattle because they better account for the environmental effects on test day, and allow for genetic differences in the shape of the lactation curve. In order to do test day model evaluations, we must know the genetic and phenotypic relationships among the test day yields. Test days close to one another are closely related. (Co)variance functions are a compact way to represent these relationships with a reduced number of parameters. This study estimated these parameters across lactation thereby providing estimates of the relationships among test day yields both within and across lactation. These estimates will be used in a test day model which will provide estimates of genetic differences in persistency, a measure of the distribution of a cows milk yield within lactation, and rate of maturity, an indication of the genetic difference between first and later lactation yield. It is computationally demanding to estimate so many relationships simultaneously. This study included milk, fat, and protein yields. This study should assist in implementation of a full test day model which will improve accuracy of genetic evaluations of milk yield.

Technical Abstract:
Co)variance components for milk, fat, and protein yields during first and second lactations were estimated from data for test days from 23,029 Holstein cows from 37 herds in Pennsylvania and Wisconsin. Four lactation stages of 75 d were defined in each lactation, and the test day nearest the center of each interval was used. A total of 9,110 observations were available for the final analysis of lactations with test days in all lactation stages. Missing values were deleted to allow a canonical transformation to be used for estimation of (co)variance matrices. Data were preadjusted for lactation curves within lactation stages using all available records. (Co)variance functions were used to describe the (co)variance structure within and across yield trait and parity. Biological functions (305-d yields, persistency, defined as difference between yields on days 280 and 60, and maturity rate, defined as difference between second dand first lactation yields) were developed from (co)variance functions. Heritabilities were between 0.09 and 0.22 for test-day yields, between 0.24 and 0.27 for 305-d yields, between 0.07 and 0.15 for persistencies and between 0.06 and 0.09 for maturity rates. Correlations between fat test-day yields in the first lactation and milk and protein yields in the second lactation were low. Phenotypic correlation of first and second lactation persistencies were low, but genetic correlations were high. Maturity rate showed correlations between 0.23 and 0.61 with 305-d yields and persistencies. Results provide an indication of the (co)variance structure within and across lactations but further research using improved strategies which can accommodate more data is needed.